When quantizing a gauge theory, we obtain spin-1 particles propagating in space-time. When we want to count the degrees of freedom of the theory or, equivalently, when we are trying to decompose the field operator in a LI basis, we use polarization vectors. My question is, in any given gauge choice, for any given gauge theory, can we choose polarization vectors however we like, as long as they form a basis?

For example, can I choose these?

\begin{equation} \epsilon_0=(1,0,0,0)\\ \epsilon_1=(0,1,0,0) \\ \epsilon_2=(0,0,1,0) \\ \epsilon_3=(0,0,0,1) \end{equation}

Or any linear combination of those? like choosing $\epsilon_{\pm}=\frac{1}{\sqrt{2}}\big(\epsilon_0 \pm\epsilon_3\big)$ which are called the forward and backward polarization vectors. I know probably some choices are smarter than others, but my question is whether we are free to choose whichever basis we want.


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